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I am trying to explicitly write out using matrices a Hamiltonian given in this condensed matter paper. In eq (3) of the paper, we have: $$ \hat{H} = a t (\tau k_x \hat{\sigma_x} + k_y \hat{\sigma_y} ) + \frac{\Delta}{2} \hat{\sigma_z} - \lambda \tau \frac{\hat{\sigma_z} - 1}{2} \hat{s_z}, $$

where $\hat{s_z}$ is the Pauli matrix for spin. $\hat{\sigma}$ denotes the Pauli matrices for the following two basis functions:

$$ | \phi_c \rangle = | d_{z^2} \rangle , | \phi_{v}^{\tau} \rangle = \frac{1}{\sqrt{2}}(| d_{x^2-y^2} \rangle + i \tau |d_{xy} \rangle) $$ Now, I am used to using Pauli matrices in the following form: $$ \hat{\sigma}_{x} = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right), \;\;\; \hat{\sigma}_{y} = \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right), \;\;\; \hat{\sigma}_{z} = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right), $$

but in the case of the given Hamiltonian, I am not sure what the 3 $\hat{\sigma}$ mean as opposed to $\hat{s_z}$, in matrix form. Could someone please point me in the right direction? Intuitively, I think that the $\hat{\sigma}$ are some sort of the 3 matrices above that have different bases, and that $\hat{s_z}$ is the third Pauli matrix shown above.

To reiterate, I am trying to identify the $2 \times 2$ matrices that correspond to the four variables: $\hat{\sigma}$ and $\hat{s_z}$.

Going off this post: $C_4$ symmetry of Chern insulator, I understand that I am dealing with pseudospins. However, the mathematical steps I must take to get what I want are not too clear to me yet.

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  • $\begingroup$ This sounds like this should be 4x4 matrices: 2 degrees of freedom for the sigma and two for the spin. $\endgroup$ – Norbert Schuch Jul 15 at 20:54
  • $\begingroup$ Are you sure the $\hat{s}_z$ is the Pauli matrix for spin? Because it usually stands for the spin matrix, i.e. $\hat{s} = \hbar/2 \hat{\sigma}$. $\endgroup$ – PhysicsTeacher Jul 15 at 23:00
  • $\begingroup$ Yes, I'm pretty sure that's a 4x4 Hamiltonian. You'll need to take a Kronecker product between the $\sigma$ and $s$ matrices. $\endgroup$ – Anyon Jul 17 at 17:56
  • $\begingroup$ Thank you all for your feedback. But I would appreciate some resource that might help me understand better, perhaps with similar expressions for 4x4 matrices. The Wikipedia page discusses the Kronecker product, but that’s for 2x2. $\endgroup$ – TribalChief Jul 17 at 21:23

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